Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({ e^{\pi i}}) ^ {11} $
Solution: Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{\pi i}}) ^ {11} = e ^ {11 \cdot (\pi i)} $ The angle of the result is $11 \cdot \pi$ , which is $11\pi$ $11\pi$ is more than $2 \pi$ . It is a common practice to keep complex number angles between $0$ and $2 \pi$ , because $e^{2 \pi i} = (e^{\pi i}) ^ 2 = (-1) ^ 2 = 1$ . We will now subtract the nearest multiple of $2 \pi$ from the angle. $ 11\pi - 10\pi = \pi $ Our result is $ e^{\pi i}$.